◀ ▲ ▶Branches / Combinatorics / Branch: Combinatorics and Discrete Mathematics

Branch: Combinatorics and Discrete Mathematics

This branch of BookofProofs is devoted to combinatorcs. Combinartorics, sometimes also called discrete mathematics, is a branch of mathematics that focusses on the study of discrete objects (as opposed to continuous ones). The main purpose is to provide a framework for counting these objects under diffent operations, including exchanges, permutations, choice situations, combinations, and many more.

Combinatorics plays a crucial role in the development of many other mathematical areas, e.g. number theory, probability theory, graph theory, geometry, and the theory of algorithms.

Theoretical minimum (in a nutshell)

As a framework for counting discrete objects, combinatorics does not require very sophisticated prerequisites to be acquainted with. However, the theoretical concepts of counting in combinatorics can become demanding for the undergraduates. The main difficulty might result not in the mere understanding of the techniques, but in the ability to recognize which technique to count things is applicable in a given situation.

Concepts you will learn in this part of BookofProofs

• Basic counting principles, e.g.
  • the Pigeonhole principle,
  • the fundamental counting principle for addition and multiplication,
  • permutations and factorials, with and without repetitions,
  • combinations and permutations of indistinguishable objects,
• You will learn problem-solving strategies for counting problems.
• Learn about some number sequences, including Stirling, Bernoulli, and about their applications.
• Learn what are recurrencies and why they are important for solving combinatorial problems and programming.
• Learn solving strategies to solve recurrences, including generating functions.

Table of Contents

1. Part: Historical Development of Combinatorics
2. Part: Set-theoretic Prerequisites Needed For Combinatorics
3. Part: Cycles, Permutations, Combinations and Variations
4. Part: Stirling Numbers
5. Proposition: Number of Relations on a Finite Set
6. Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters
7. Part: Solving Strategies and Sample Solutions to Problems in Combinatorics
8. Part: Discrete Calculus and Difference Equations
9. Proposition: Multinomial Coefficient

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